Elements of Algebra

488 ADDITIONS. CHAP. I. were exactly 6 hours, it would make one day at the end of four common years: but if we wish to know, exactly, at the end of how many years this difference will produce a certain number of days, we must seek the ratio between $24^h$, and $5^h. 48'. 49''$, which we find to be $\frac{86400}{20929}$; so that at the end of $86400$ common years, we must intercalate $20929$ days, in order to reduce them to tropical years. Now, as the ratio of $86400$ to $20929$ is expressed in very high terms, let it be required to find ratios, in lower terms, as near this as possible. For this purpose, we must reduce the fraction $\frac{86400}{20929}$ to a continued fraction, by the rule given in Art. 4, which is the same as that by which the greatest common divisor of two given numbers is found. This will give us $20929)86400(4 = \alpha$ $83716$ $2684)20929(7 = \beta$ $18788$ $2141)2684(1 = \gamma$ $2141$ $543)2141(3 = \delta$ $1629$ $512)543(1 = \varepsilon$ $512$ $31)512(16 = \zeta$ $496$ $16)31(1 = \eta$ $16$ $15)16(1 = \theta$ $15$ $1)15(15 = \iota$ $15$ $0.$ Now, as we know all the quotients $\alpha, \beta, \gamma, \ldots$ we easily form from them the series $\frac{A}{A'}, \frac{B}{B'}, \ldots$ in the following